Positivity-preserving C2 rational cubic spline

R ESEARCH
ARTICLE
doi: 10.2306/scienceasia1513-1874.2013.39.208
ScienceAsia 39 (2013): 208–213
Positivity-preserving C 2 rational cubic spline
interpolation
Muhammad Abbasa,∗ , Ahmad Abd Majida , Mohd Nain Hj Awangb , Jamaludin Md Alia
a
b
∗
School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang Malaysia
School of Distance Education, Universiti Sains Malaysia, 11800 Penang Malaysia
Corresponding author, e-mail: [email protected]
Received 5 Jun 2012
Accepted 20 Mar 2013
ABSTRACT: This work addresses the shape preserving interpolation problem for visualization of positive data. A piecewise
rational function in cubic/quadratic form involving three shape parameters is presented. Simple data dependent conditions
for a single shape parameter are derived to preserve the inherited shape feature (positivity) of data. The remaining two shape
parameters are left free for the designer to modify the shape of positive curves as per industrial needs. The interpolant is
not only C 2 , local, computationally economical, but it is also a visually pleasant and smooth in comparison with existing
schemes. Several numerical examples are supplied to illustrate the proposed interpolant.
KEYWORDS: shape preserving interpolation, data visualization, positive data, shape parameters, parametric continuity
INTRODUCTION
The problem of constructing a shape preserving curve
through given data points is one of the basic problems
in computer graphics, computer aided geometric design, data visualization and engineering. Curve design
plays a significant role not only in these fields but
also in manufacturing different products such as ship
design, car modelling, and aeroplane fuselages and
wings.
A solution to the aforementioned problem results
in the construction of some interpolants that preserve
the inherited shape features of data such as positivity,
monotonicity, and convexity. The goal of this paper
is to preserve the genetic characteristic (positivity)
of data. The positivity-preserving problem occurs in
visualizing a physical quantity that cannot be negative
which may arise if the data is taken from some
scientific, social, or business environments.
Classical methods, with the polynomial spline
functions, show smooth and visually pleasing results
but usually ignore this shape feature of data and
thus yield solutions exhibiting undesirable undulations or oscillations. Piecewise cubic Hermite interpolating polynomial (a built-in MATLAB program
called PCHIP) has the ability to remove the undesired
undulations but the shape preserving visual model
depicts the tight display of data.
A great deal of research on this topic has been
done, especially on the shape preserving interpolations. Some C 1 piecewise rational cubic interwww.scienceasia.org
polants 1–5 have a common feature in a way that no additional knots are used for shape preservation of positive 2D and 3D data. In contrast, the cubic Hermite
interpolation 6 and cubic polynomial spline 7 preserve
the shape of data by inserting one or two additional
knots in the subinterval where the interpolants do not
attain the desired shape characteristics of data. Some
rational cubic interpolations 8, 9 have been developed
for value control, inflection-point control, convexity
control of the interpolation at a point, and constrained
curve control which were based on function values.
The authors assumed suitable values of parameters to
achieve a C 2 continuous curve and the methods work
for only equally spaced data. Costantini 10 solved
the shape preserving boundary valued problems using
polynomial spline interpolation with arbitrary constraints.
The problem of shape preserving interpolants for
visualization of positive, monotone and convex data
has been solved using a C 2 rational cubic function with shape parameters 11, 12 . Simple data dependent constraints for free parameters were derived
to achieve the desired shape features of the data.
The schemes 4, 12 did not provide the liberty to the
designer for the refinement of shape of curves as per
industrial needs. Lamberti and Manni 13 presented and
investigated the approximation order of a global C 2
shape preserving interpolating function using parametric cubic curves. The tension parameters were
used to control the shape of the curve. The authors
derived the necessary and sufficient conditions for
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ScienceAsia 39 (2013)
convexity but only sufficient conditions for positivity
and monotonicity of data.
In this paper, a C 2 piecewise rational cubic spline
scheme with three shape parameters is developed
to address the problem of constructing a positivitypreserving curve through positive data. It improves on
the existing schemes in following ways. (1) In Refs.
1, 4–6, the smoothness of interpolants is C 1 while in
this work it is C 2 . (2) No additional points (knots)
are inserted in the proposed interpolant. In contrast,
the methods in Refs. 6, 7 achieve the desired shape
of curve by inserting of additional knots in the subinterval where the interpolant loses positivity. (3) The
proposed scheme has been verified through several
numerical examples and it is found to be local in
comparison with global schemes 13 , computationally
economical, and produces smoother graphical results
than existing schemes 1, 4–6, 11, 12 . (4) The authors in
Ref. 7 achieved the values of derivative parameters
by solving the three systems of linear equations,
which is computationally expensive as compared to
method developed in this paper where there exists
only one tridiagonal system of linear equations for
finding the values of derivative parameters. (5) The
schemes developed in Refs. 8, 9 work for only equally
spaced data while the scheme developed in this paper
works for both equally and unequally spaced data.
(6) The authors in Ref. 9 assumed certain function
values and derivative values to control the shape of
the data while in this paper, simple data dependent
constraints for single shape parameters are derived to
achieve a positivity-preserving curve through positive
data. (7) The proposed curve scheme is unique in its
representation and it is equally applicable for the data
with derivative or without derivatives.
where S 0 (x) and S 00 (x) denote the first and second
order derivatives with respect to x, and the + and
− subscripts denote the right and left derivatives, respectively. From (2), the C 2 interpolating conditions
produce the following unknown coefficients ξi , i =
0, 1, 2, 3:
ξ0
ξ1
ξ2
ξ3
= ui fi
= fi (2ui + vi + wi ) + ui hi di
= fi+1 (ui + 2vi + wi ) − vi hi di+1
= vi fi+1
(3)
where hi = xi+1 − xi , θ = (x − xi )/hi , θ ∈ [0, 1]
and ui , vi > 0, wi > 0 are shape parameters that are
used to control the shape of interpolating curve. Let
di denote the derivative values at given knots xi that
are used for the smoothness of curve.
The C 2 piecewise rational cubic function (1) is
reformulated after using (3):
S(xi ) =
pi (θ)
qi (θ)
(4)
with,
pi (θ) = ui fi (1 − θ)3
+ (fi (2ui + vi + wi ) + ui hi di ) θ(1 − θ)2
+ (fi+1 (ui + 2vi + wi ) − vi hi di+1 ) θ2 (1 − θ)
+ vi fi+1 θ3
qi (θ) = (1 − θ)2 ui + θ(1 − θ)(wi + ui + vi ) + θ2 vi .
The C 2 interpolating conditions (2) produce the following system of linear equations for the computation
of derivatives parameters di .
αi di−1 + δi di + γi di+1 = λi
(5)
RATIONAL CUBIC SPLINE FUNCTION
Let {(xi , fi ) : i = 0, 1, 2, . . ., n} be the given set of
data points such that x0 < x1 < x2 < . . . < xn . A
piecewise rational cubic function 14 with three shape
parameters in each subinterval Ii = [xi , xi+1 ], i =
0, 1, 2, . . ., n − 1 is defined as:
P3
S(x) ≡ Si (x) =
− θ)3−i θi ξi
qi (θ)
i=0 (1
(1)
αi = ui ui−1 hi
δi = hi ui (ui−1 + vi−1 + wi−1 )
+hi−1 vi−1 (ui + vi + wi )
γi = vi vi−1 hi−1
λi = vi−1 hi−1 (ui + 2vi + wi )∆i
+ui hi (2ui−1 + vi−1 + wi−1 )∆i−1
where ∆i = (fi+1 − fi )/.hi .
The following interpolatory conditions are imposed on
the piecewise rational cubic function (1) for the C 2
continuity:
S(xi ) = fi ,
S(xi+1 ) = fi+1 ,
S 0 (xi ) = di ,
S 0 (xi+1 ) = di+1 ,
00
00
S (xi+ ) = S (xi− ), i = 1, 2, . . ., n − 1,
with,
(2)
Remark 1 The system of linear equations defined in
(5) is strictly tridiagonal and has a unique solution for
the derivatives parameters di , i = 1, 2, . . ., n − 1 for
all ui , vi > 0, and wi > 0. Moreover, it is efficient
to apply LU decomposition to solve the system for the
0
values of derivatives parameters di s.
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ScienceAsia 39 (2013)
Remark 2 To make the rational cubic function
smoother, C 2 continuity is applied at each knot. The
system (5) involves n − 1 linear equations. There are
n + 1 unknown derivative values. Hence two more
equations are required for unique solution. For this,
we impose end conditions at end knots as:
S 0 (x0 ) = d0
(6)
S 0 (xn ) = dn
(7)
Remark 3 For the values of shape parameters set as
ui = 1, vi = 1 and wi = 0 in each subinterval Ii =
[xi , xi+1 ], i = 0, 1, 2, . . ., n − 1, the rational cubic
function reduces to the standard cubic Hermite spline.
SHAPE PRESERVING POSITIVE C 2
RATIONAL CUBIC CURVE INTERPOLATION
In this section, we discuss the problem of getting
a positivity-preserving curve by using a C 2 rational
cubic spline (4). Simple data dependent conditions
for a single shape parameter are derived to maintain
the natural characteristic of positive data.
Let {(xi , fi ) : i = 0, 1, 2, . . ., n} such that x0 <
x1 < x2 < . . . < xn be a given positive set of data if
fi > 0,
∀i
(8)
The rational cubic function (4) preserves the positivity
of positive data if both pi (θ) > 0 and qi (θ) > 0.
Therefore the necessary conditions ui > 0, vi > 0
and wi > 0 are enough for qi (θ) > 0. According to
Ref. 15, pi (θ) > 0 if
(p0i (0), p0i (1)) ∈ R1 ∪ R2
where,
This yields the following constraints for the shape
parameter wi :
−ui (2fi + hi di )
− vi ,
fi
vi (hi di+1 − 2fi+1 )
wi >
− ui
fi+1
wi >
(9)
(10)
Furthermore, (p0i (0), p0i (1)) ∈ R2 if
36fi fi+1 (a2 + b2 + ab − 3∆i (a + b) + 3∆2i )
+ 4hi (a3 fi+1 − b3 fi ) + 3(afi+1 − bfi )
(2hi ab − 3afi+1 + 3bfi ) − h2i a2 b2 > 0
(11)
where a = p0i (0) and b = p0i (1). The constraints on
wi can also be computed from (11) but an efficient
and alternative choice is to use the constraints given
in (9)–(10) for positivity-preserving graphical results.
The above results can be summarized as follows.
Theorem 1 The C 2 piecewise rational cubic function
(4) preserves the positivity of positive data in each
subinterval [xi , xi+1 ] if the shape parameters satisfy
the following conditions:
ui > 0, vi >
( 0 −u (2f +h d )
)
0, i fii i i − vi ,
wi > max
vi (hi di+1 −2fi+1 )
− ui
fi+1
The above result can be rewritten as:
(
)
0, −ui (2ffii+hi di ) − vi ,
wi = li + max
, li > 0
vi (hi di+1 −2fi+1 )
− ui
fi+1
NUMERICAL EXAMPLES
R1 =
(a, b) : a >
3pi (1)
−3pi (0)
, b<
hi
hi

(a, b) : 36fi fi+1 (a2 + b2 + ab



−3∆i (a + b) + 3∆2i ) + 4hi (a3 fi+1 − b3 fi )
R2 =
+3(afi+1 − bfi )(2hi ab − 3afi+1 + 3bfi )



−h2i a2 b2 > 0
It can easily be shown that
(wi + vi )fi + ui (hi di − fi )
hi
v
(h
d
+
f
i
i
i+1
i+1 ) − (wi + ui )fi+1
p0 (1) =
hi
p0 (0) =
Now (p0i (0), p0i (1)) ∈ R1 if
p0i (0) >
www.scienceasia.org
−3pi (0)
,
hi
p0i (1) <
3pi (1)
hi
In this section, the efficiency of the proposed
positivity-preserving scheme through several numerical examples is presented. A comparison of C 2
scheme with PCHIP and existing schemes are also part
of this section.
Example 1 The cubic Hermite spline scheme in Ref.
16 and PCHIP have been used to produce curves
in Fig. 1a,b, respectively, through positive data taken
in Table 1 which is borrowed from Ref. 11. These
figures depict the flaws and tightness of these splines.
The efficiency of the proposed scheme can be seen in
Fig. 1c,d. A remarkable difference in the smoothness
with a pleasant graphical view is visible in Fig. 1c,d,
drawn by the proposed C 2 positive rational cubic
spline scheme. Table 2 gives the computed values
from the developed scheme for Fig. 1d.
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ScienceAsia 39 (2013)
Table 3 A positive data set.
Table 1 A positive data set.
xi
fi
0.0
0.25
2
3
1.0
1.0
1.71
11.10
4
i
xi
fi
1.80
25
1
2
3
4
5
6
7
1
2
4
5
7
8
9
24.6162
2.4616
41.0270
4.1027
57.4378
5.7438
0.5744
Table 2 Numerical results.
i
1
2
−7.296
0.75
2.25
2.25
0.068
di
∆i
ui
vi
wi
3
3.079
14.429
2.25
2.25
0.004
4
69.242
139
2.25
2.25
0.001
154.570
...
...
...
...
Example 2 A positive data set taken in Table 3 which
is borrowed from Ref. 5. The data was collected from
a chemical experiment, where the molar volume of
the gas (in l/mol) at different temperature codes was
observed. The x-values are temperature codes and
the f -values are the molar volumes of the gas. The
curve in Fig. 2a is drawn by the cubic Hermite spline
scheme 16 without constraints of positivity. Generally,
the curve is smooth, but the positivity is lost which
does not make any sense physically. On the other
hand, the two curves in Fig. 2c,d are generated by the
proposed positivity-preserving rational cubic function
with different values of shape parameters. It is noted
25
(a)
20
di
1
2
3
4
5
6
7
−35.967
−0.7612
−18.973
2.8783
−0.1731
−138.23
18.093
0
0
1.5
0.5
xi
1
1.5
xi
10
5
5
0.5
1
xi
1.5
0
0
60
(b)
20
10
0
−10
0
5
Temperature Code
50
40
30
20
10
0
0
10
0.5
1
1.5
xi
Fig. 1 Comparison between (a) cubic Hermite spline curve
(b) PCHIP curve and C 2 positive rational cubic curve with
values of shape parameters set as: (c) ui = vi = 0.5;
(d) ui = vi = 2.25.
10
5
Temperature Code
10
(d)
50
40
30
20
10
0
0
5
Temperature Code
60
(c)
f
10
0.005
0.005
0.005
0.005
0.005
14.2540
...
30
Molal Volume of gas
15
1.5
1.5
1.5
1.5
1.5
1.5
...
60
i
15
fi
20
1.5
1.5
1.5
1.5
1.5
1.5
...
40
(d)
20
−22.131
19.283
−36.924
26.668
−51.694
−5.1694
...
50
25
(c)
wi
Example 3 A positive data set in Table 5 gives the
velocity of wind. The velocity is inherently positive
Molal Volume of gas
i
f
fi
5
1
vi
(a)
0
0.5
ui
60
15
−10
0
∆i
that proposed positivity-preserving curves (Fig. 2c,d)
are more visually pleasing and smooth as compared
to PCHIP curve (Fig. 2b). Table 4 demonstrates the
computed values from developed scheme of Fig. 2d.
(b)
10
0
0
i
20
10
25
Table 4 Numerical results.
Molal Volume of gas
1
Molal Volume of gas
i
5
Temperature Code
10
50
40
30
20
10
0
0
Fig. 2 Comparison between (a) cubic Hermite Spline curve
(b) PCHIP curve and positivity-preserving C 2 rational cubic
curve with values of shape parameters set as: (c) ui = vi =
0.25; (d) ui = vi = 1.5.
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ScienceAsia 39 (2013)
Table 5 Positive data points.
2.0
0.6
0.1
0.13
1.0
0.5
1.1
0.25
0.2
di
1
2
3
4
5
6
7
8
9
−7.4000
−3.5912
−1.0353
1.5744
0.1377
0.0947
−0.0832
−1.9277
1.05
∆i
ui
vi
wi
-5.6
2.0
0.06
1.74
−1.0
1.2
−1.7
−0.05
...
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
...
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
...
0.005
7.9468
0.005
0.005
0.005
0.005
0.005
14.2820
...
and we therefore, require a C 2 rational cubic function
with shape parameters to preserve this shape characteristic. The x-values are time (min) and the f values are velocity of wind (km/min). In Fig. 3a,b the
Hermite curve does not maintain the positivity through
positive data and the PCHIP curve depicts the tightness of the positive curve, respectively. To overcome
these flaws, the curves in Fig. 3c,d are produced by a
positivity-preserving C 2 rational cubic function which
preserves the positivity of data everywhere. These
rational curves are more visually pleasing and smooth
as compared to the PCHIP curve. Table 6 represents
the computed values from the developed scheme of
Fig. 3d.
Example 4 The cubic Hermite spline scheme 16 and
PCHIP have been used to produce Fig. 4a,b, respectively, through positive data taken in Table 7 which
is borrowed from Ref. 12. These curves depict
the flaws (negativity) and tightness of these splines.
The efficiency of proposed scheme can be seen in
Fig. 4c,d. A remarkable difference in the smoothness
with a pleasant graphical view is visible in Fig. 4b and
Fig. 4c,d, drawn by PCHIP curve scheme and proposed C 2 rational cubic spline scheme, respectively.
Table 8 demonstrates the computed values from the
developed scheme of Fig. 4d.
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1
0.5
0
−0.5
0
1
2
3
Time (min)
(b)
1.5
1
0.5
0
0
4
2
Table 6 Numerical results.
i
Velocity of wind (km/min)
0
0.25
0.5
1.0
1.5
2.0
2.5
3.0
4.0
1.5
1.5
1
0.5
0
0
1
2
3
Time (min)
4
2
(c)
1
2
3
Time (min)
Velocity of wind (km/min)
1
2
3
4
5
6
7
8
9
fi
Velocity of wind (km/min)
xi
2
(a)
Velocity of wind (km/min)
i
2
(d)
1.5
1
0.5
0
0
4
1
2
3
Time (min)
4
Fig. 3 Comparison of (a) Cubic Hermite spline curve
(b) PCHIP curve with positivity-preserving rational cubic
curve with shape parameters (c) ui = vi = 0.25; (d) ui =
vi = 2.5.
Table 7 Positive data points.
i
xi
1
2
3
4
5
6
7
2.0
3.0
7.0
8.0
9.0
13.0
14.0
fi
10.0
2.0
3.0
7.0
2.0
3.0
10.0
CONCLUDING REMARKS
A C 2 rational cubic function with three shape parameters has been constructed in this paper to preserve the
positivity of a curve through positive regular data. Examples suggest that the proposed interpolant appears
to produce smoother graphical results.
Table 8 Numerical results.
i
di
1
2
3
4
5
6
7
−9.65
−2.7481
3.6429
1.7013
−5.7728
9.446
8.350
∆i
ui
vi
wi
-8
2.25
4.0
−5.0
0.25
7.0
...
1.5
1.5
1.5
1.5
1.5
1.5
...
1.5
1.5
1.5
1.5
1.5
1.5
...
1.500
0.0097
0.001
0.001
0.0249
0.001
...
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ScienceAsia 39 (2013)
10
10
(a)
(b)
8
7.
fi
fi
5
0
2
6
8.
4
4
6
8
xi
10
12
2
2
14
4
6
8
xi
10
12
14
9.
10
10
(c)
(d)
6
6
10.
fi
8
fi
8
4
4
2
2
0
2
4
6
8
xi
10
12
14
0
2
11.
4
6
8
xi
10
12
14
Fig. 4 Comparison of (a) cubic Hermite Spline curve
(b) PCHIP curve with positivity-preserving rational cubic
curve with values of shape parameters (c) ui = vi = 0.25;
(d) ui = vi = 2.5.
In the future, we will try to extend the C 2 rational
cubic function to rational bi-cubic functions and bicubic partially blended rational functions to solve the
C 2 shape preserving interpolating surface problem in
the rectangular case.
12.
13.
14.
15.
16.
Acknowledgements: The authors are really grateful to
the anonymous referees for their inspiring comments which
helped in improving this manuscript significantly. They are
also pleased to acknowledge the financial support of the
School of Mathematical Sciences, Universiti Sains Malaysia
and the University of Sargodha, Pakistan.
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